Basically the title: the phisycs of juggling a ping-pong ball.pdf

1. The physics of juggling a spinning ping-pong ball
Ralf Widenhorn
Citation: Am. J. Phys. 84, (2016); doi: 10.1119/1.4964104
View online: http://dx.doi.org/10.1119/1.4964104
View Table of Contents: http://aapt.scitation.org/toc/ajp/84/12
Published by the American Association of Physics Teachers
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2. The physics of juggling a spinning ping-pong ball
Ralf Widenhorna)
Department of Physics, Portland State University, Portland, Oregon 97201
(Received 14 August 2015; accepted 18 September 2016)
Juggling a spinning ball with a ping-pong paddle represents a challenge both in terms of hand-eye
coordination and physics concepts. Here, we analyze the ping-pong ball’s motion, and explore how
the correct paddle angle relates to the ball’s spin and speed, as it moves vertically up and down. For
students, this requires engaging with concepts like momentum, angular momentum, free-body
diagrams, and friction. The activities described in this article include high-speed video motion
tracking of the ping-pong ball and the investigation of the frictional characteristics of the paddle. They
can be done in a physics lab or at home, requiring only inexpensive or commonly used equipment,
and can be undertaken by high school or college students.V
C 2016 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4964104]
I. INTRODUCTION
Students usually get their first introduction to physics
through mechanics. The study of motion provides various
opportunities for lab activities. Although students have devel-
oped an intuition through everyday experience of how objects
move, the challenges for students to correctly understand these
concepts have been well documented since the early years
of physics education research.1
While a standard laboratory
experiment aims to teach important concepts and experimental
skills, we find few “typical” experiments excite our students.
Furthermore, labs frequently suffer from being “cookbook
style,” with little room for students to actively engage and
explore physics phenomena, or to develop true experimental
skills that have been identified as important by the American
Association of Physics Teachers Recommendations for the
Undergraduate Physics Laboratory Curriculum2
and the Next
Generation Science Standards.3
One way to allow students to
more freely explore basic physics principles and develop
experimental skills is to have them work on a project lab.4
A
good project leaves room for students to explore different
aspects of a phenomenon, is challenging, captures a student’s
interest, and at the same time provides the opportunity to dis-
cuss experimental design.
The exploration of physics phenomena in sports can pro-
vide an extra stimulus to spark the interest of many students
and has been the subject of several textbooks.5,6
However, it
is difficult to investigate sports activities in a lab environ-
ment. Table tennis, often referred to colloquially as ping-
pong, uses a light ball that can be easily studied in a confined
space. A key difference between competitive table tennis
and recreational ping-pong is the use of spin. The spin of a
ping-pong ball is difficult to observe directly, but its effect
on all aspects of the game is profound. The mass of the ball
is small and the ball’s trajectory and its motion upon bounc-
ing off the table and paddle are non-intuitive for all but the
most experienced players (see Fig. 1). While this makes it
more difficult to predict a ball’s trajectory, it also makes the
motion more intriguing to analyze. Concepts like kinematics,
projectile motion, free-body diagrams, friction, air resis-
tance, the Magnus force, kinetic energy, rotational kinetic
energy, impulse, forces, and angular momentum can all play
an important role in such an analysis. Various articles have
been written on the bounce of spinning balls in ping-
pong7–12
and other types of spinning balls upon hitting a rac-
quet, paddle, club, or the ground.13–25
Following up on these
studies, we present a project that many students who enjoy
ball sports will find to be a challenge to their hand-eye coor-
dination and their physics skills. For our study we will focus
on the effect of spin on the bounce of the ping-pong ball.
Due to the relatively small speeds involved we will neglect
drag forces26–28
and the curving of the ball due to the
Magnus force.29–33
The goal here is to hit a ping-pong ball upward with some
spin, and try to control it when it impacts the paddle to send
it straight up again, so that it can easily be caught afterwards.
Such an exercise helps a player to get a feel for the speed
and tackiness of the paddle. For the study described in this
manuscript, we use a standard 40-mm plastic ping-pong ball
with a pre-assembled entry level Stiga Inspire paddle with
Magic rubber (1.5-mm sponge)34
and the competitive grade
combo of a Butterfly Tenergy 05 rubber (2.1-mm sponge)35
on a Timo Boll Spirit blade.36
The lower quality Magic rub-
ber had lost its initial tackiness while the Tenergy rubber
was still tacky.
One can send a ball without spin straight up, by placing
the paddle flat under the ball, but in the case of a spinning
ball the paddle needs to be angled as shown in Fig. 2. With
some practice, one can develop a good intuition on how to
angle the paddle and juggle the spinning ball multiple times
by alternating the angle of the paddle from being titled
clockwise to counterclockwise, sending the ball straight up
each time. To analyze this motion a couple of research ques-
tions might include: “At what angle a, does the paddle need
to be placed to have the ball go vertically upward for differ-
ent initial and final speeds and spins of the ball?” and “How
does this angle depend on the type of paddle?”
II. MOTION TRACKING
We used a point-and-shoot Casio Exilim EX-FH100
camera37
that captures video in the high frame rate shooting
modes of 240 fps at 448 336 pixels, and at 1,000 fps at
224 64 pixels. The compact consumer camera was equipped
with an SD card rated for 10MB/s and was mounted on a stan-
dard tripod. We use Vernier Logger Pro software to extract
data from the videos using frame-by-frame tracking of the
ball.38
For students new to ping-pong, the first step is to practice
how to brush the ping-pong ball from underneath with the
paddle such that the ball has a large spin and flies up approx-
imately vertically. Next, students will record their attempts
936 Am. J. Phys. 84 (12), December 2016 http://aapt.org/ajp V
C 2016 American Association of Physics Teachers 936

3. to hold the paddle such that the ball pops straight up, catch-
ing the ball after each try. Even though it is a fun challenge,
one does not need to juggle the ball multiple times for this
project. For the video capture, it is easiest to use a tripod so
that one can easily adjust the level and angle of the camera.
For the study described in this section, we chose a frame rate
of 240 fps, which provides sufficient spatial resolution for
motion tracking. One finds that the ball velocities for this
experiment are on the order of a few meters per second or
less and are easily tracked at a lower frame rate. However,
the spin can be such that a high frame rate is required to
track the rotational speed of the ball. The 240 fps frame rate
would have a ball spinning at 60 rev/s rotate by an easily
traceable quarter of revolution from frame to frame.
Particularly for students interested in engineering, this would
be a good opportunity to explore signal processing, the
Nyquist frequency, and aliasing effects. Drawing a black line
around the circumference of the ball and adding two dots at
the ball’s polar opposites allowed the ability to track the
ball’s spin. The rotational velocity is found by counting the
rotations of the line or dots from frame to frame. Video
acquisition should be done in a well-lit room, avoiding
incandescence and other light sources that exhibit power line
flicker.
Figure 1 shows the impact of the spin on the bounce of the
ball. The ball rotated counterclockwise (from the perspective
of the reader) three times over 36 frames before bouncing
off the paddle. Defining a clockwise rotation as positive,
the angular velocity of the spinning ball was therefore xi
¼ 240 fps=ð12 f=revÞ ¼ 20 rev=s. Following the bounce,
the ball rotated clockwise once over 38 frames, resulting in
an angular velocity of xf ¼ 6:3 rev=s.
To prevent the ball from bouncing off to the side, one
needs to angle the paddle. The counterclockwise rotation of
the ball before impact, as shown in Fig. 2(a), requires
angling the paddle as depicted in Fig. 2(b). Moving the pad-
dle as indicated with the arrows in Figs. 2(b) and 2(e) will
add extra speed and counter-rotation to the ball. Upon
impact, the ball rises and then falls, as shown in Figs. 2(c)
and 2(d) while now spinning clockwise. A ball with this spin
would jump to the right if the paddle is held horizontally,
hence the paddle needs to be angled as shown in Fig. 2(d).
Fig. 1. Overlay image of a spinning ball dropping vertically onto a horizon-
tal paddle. The video was taken at a frame rate of 240 fps, and the ball loca-
tion is shown for every tenth frame, approximately 41.7 ms apart.
Fig. 2. Sequence of images when juggling a spinning ping-pong ball. The frames were selected to illustrate one complete juggling cycle: (a) ball falling verti-
cally and rotating counterclockwise; (b) angled paddle impacting the ball, the paddle moves in the direction as indicated by the arrow; (c) ball moving verti-
cally upwards after impact while rotating clockwise; (d) ball dropping back vertically toward the paddle while still rotating clockwise; (e) angled paddle
impacting the ball, the paddle moves in the direction as indicated by the arrow; (f) ball moving vertically upward after impact while rotating counterclockwise.
937 Am. J. Phys., Vol. 84, No. 12, December 2016 Ralf Widenhorn 937

4. After impact, the ball will move upwards with a counter-
clockwise spin as shown in Fig. 2(e). A skilled student can
repeat the sequence as the ball falls again, as in Fig. 2(a). For
the motion analysis, it is sufficient to consider the first three
figures as the paddle angle and ball motion of the following
part of the sequence are symmetric with rotational directions
and the horizontal axis flipped.
To convert the pixel distance in the frame-by-frame track-
ing of the ball to a physical distance, one needs to calibrate it
with an object of known length. The calibration was done by
displaying a 50-cm ruler in the plane of motion at one point
during the video capture. With this reference distance and
the frame rate of the video capture, one can plot horizontal
and vertical distances and velocities as a function of time.
For this study, the camera was about 3 m away from the ball,
resulting in small angles for the vertical positions analyzed.
Figure 3(a) shows the position during the fall of a ball
from close to its peak motion to after it hits the paddle. The
ball drops and rises almost vertically with a slight movement
to the right throughout its trajectory. The vertical position
before impact varies quadratically as a function of time. The
best-fit line results in a gravitational acceleration slightly
larger than the theoretical value, pointing to the calibration
length being slightly off. This can be due the fact that the
height of the exact contact point with the ball, and the height
from which the ball dropped, varied from trial to trial and
therefore is not always in line with the position of the ruler
during calibration. The exact distance of the plane of motion
of the ball from the camera varied from trial to trial as
well. Each point on the velocity versus time graph shown in
Fig. 3(b) is calculated from a seven frames kernel to calcu-
late the derivative of the position data in Fig. 3(a). This
causes the smoothing of the velocity data, which is espe-
cially visible for vy around the point of impact.
The data shown in Fig. 3 indicate that the horizontal
velocity was small throughout the motion, and the ball suc-
cessfully bounces almost vertically off the paddle. Under the
influence of gravity, the magnitude of the vertical velocity
increases linearly with time until the ball hits the paddle and
changes direction. The large magnitude of the velocity after
impact indicates that the paddle added extra translational
kinetic energy to the ball. The velocity vi before and the
velocity vf after impact can be obtained from linear fits to the
corresponding data points. The data points for the three
frames before and after the bounce are not accurate, due to
the smoothing of the velocity data and are not included in
the fits. The small changes in the horizontal velocity further
indicate that the speed of the ball is small enough so that the
Magnus force did not have a significant impact on the
trajectory.
III. MOTION ANALYSIS
To analyze the motion of the ball, one needs to consider
both its spin and linear velocity. Figure 4(a) shows all forces
acting on the ball on impact. The weight of the ball is
included for pedagogical reasons though for most cases it
will be small compared to the other forces during impact.
Choosing the center of the ping-pong ball as the rotational
axis, we can calculate the change in angular momentum by
multiplying the torque s by the time over which it acts Dt,
giving
DL ¼ sDt ¼ Ff rDt ¼ FrDt sin a; (1)
where r is the radius of the ball (see Fig. 4 for the definitions
of Ff, F, and a). Meanwhile, the change in linear momentum
in the vertical direction is
Dp ¼ FDt FgDt; (2)
which can be expressed as
Fig. 3. (a) Frame-by-frame position tracking of the ball in the horizontal and
vertical directions. (b) Corresponding horizontal and vertical velocities as a
function of time.
Fig. 4. (a) Forces acting on the ball as it bounces off the paddle upon impact.
(b) The vector sum of the frictional force Ff and the normal force FN result in a
vertical net force on the ball. (c) Impulse and change in momentum of the ball.
938 Am. J. Phys., Vol. 84, No. 12, December 2016 Ralf Widenhorn 938

5. F ¼
Dp
Dt
þ Fg: (3)
Inserting Eq. (3) into Eq. (1) then gives
DL ¼
Dp
Dt
þ Fg
rDt sin a; (4)
and solving for a results in
sin a ¼
DL
rDt Dp=Dt þ Fg
: (5)
Since the weight of the ping-pong ball is small and in most
cases
Dp=Dt Fg; (6)
this result becomes
sin a ¼
DL
rDp
: (7)
We note that in vector notation this equation can be
expressed simply as D~
L ¼ ~
r D~
p. To find the angle a in
terms of the measurable quantities Dx and Dv, we need to
replace Dp and DL in Eq. (7). Though one could include the
thickness of the ping-ping ball shell,39,40
we are assuming
the ping-pong ball has the moment of inertia of a hollow
sphere so the change in angular momentum can be calculated
using
I ¼
2
3
mr2
; (8)
giving
DL ¼ I xf xi
ð Þ ¼
2
3
mr2
Dx: (9)
Using
Dp ¼ mðvf viÞ ¼ mDv (10)
and inserting Eqs. (9) and (10) into Eq. (7) results in
sin a ¼
2rDx
3Dv
: (11)
The minimum required coefficient of friction of the rubber
sheet at the angle a can then be calculated (see Fig. 4) from
lmin ¼
Ff
FN
¼ tan a: (12)
Table I shows a set of data taken with both paddles. Trials 1
and 2 attempted to move the paddle very little on impact and
still have the ball bounce upward. One can observe that for
both paddles, the ball bounces off with a slightly smaller
speed. The elastic Tenergy rubber sheet reverses the spin
almost completely. Other trials, with little movement of the
paddle, showed xf is generally slightly smaller than xi, but
overall confirmed that most of the spin is inverted and there-
fore has a large tangential coefficient of restitution (ratio of the
outgoing and incoming velocity tangential to the ball surface)
for this rubber sheet. For the Magic rubber, there is almost no
spin after the bounce. It was generally found that little paddle
movement resulted in a small inverted spin corresponding to a
tangential coefficient of restitution of close to zero for this rub-
ber sheet. The angle a is calculated using Eq. (11) and com-
pared with ameasured, which is determined by measuring the
physical placement of the paddle in the frame of impact using
a virtual ruler in Logger Pro.
If we want to send a ball with xi and vi vertically upwards
to the same level, with little movement of the paddle, we can
determine the angle at which to place the paddle: Dv needs to
be equal to 2vi to reach the same level. For the Tenergy rub-
ber, we can approximate Dx ¼ 2xi and Eq. (11) results in
sin a ¼
2rxi
3vi
: (13)
Meanwhile, in a first approximation, the ball loses most of
its rotation upon impact for the Magic rubber sheet.
Assuming xf ¼ 0 Eq. (11) leads to
sin a ¼
rxi
3vi
: (14)
Sending the ball back to the same level with little movement
of the paddle therefore requires angling the tacky and elastic
Tenergy rubber paddle at a larger angle than the paddle with
the Magic rubber. For the Tenergy paddle, the rotation is
inverted and one could juggle a ball as often as one likes up
and down by alternating the paddle angle from þa to a.
The Magic rubber causes the ball to lose most of its spin,
and one could place the paddle almost horizontally on the
next stroke.
For both paddles, we can vary the spin and velocity of the
ball by striking it with a greater paddle speed. Depending on
the direction of the paddle motion, one imparts more spin or
translational velocity on the ball. As long as Eq. (11) is satis-
fied, the ball will travel straight up. By increasing the spin,
one increases the change in angular momentum and hence
the paddle angle. Trials 3 and 4 both increase the spin and
translational velocity. A large change in the angular velocity,
as in trial 4, will result in a more angled paddle. Trial 5 is an
example of a slow ball being sent back to roughly the same
level with a slight increase in rotational kinetic energy and a
strongly angled paddle. Trial 6 shows that one can invert the
spin with the Magic rubber paddle, however, this requires
moving the paddle quickly in the direction of the angle a.
The result for the Magic rubber demonstrates that one can
continuously juggle the ball if one adds a significant swing
Table I. Paddle angle for different vertical and angular velocities. The quan-
tities xi, xf, vi, and vf are determined from motion tracking, ameasured is found
from position measurements on the frame of impact, a is calculated using
Eq. (11), and lmin is found from Eq. (12).
Trial Rubber
xi
(rev/s)
xf
(rev/s)
vi
(m/s)
vf
(m/s)
a
(deg)
ameasured
(deg)
%
diff.
lmin
¼ tan a
1 Tenergy 17 17 3.7 3.2 24 20 16 0.44
2 Magic 18 2 2.4 2.2 20 19 8 0.37
3 Tenergy 12 37 3.2 4.1 34 34 0 0.67
4 Tenergy 28 41 2.4 5 51 52 2 1.22
5 Tenergy 18 25 2.1 2.2 54 45 17 1.37
6 Magic 12 11 2.1 2.6 25 24
3 0.46
939 Am. J. Phys., Vol. 84, No. 12, December 2016 Ralf Widenhorn 939

6. of the paddle. Thus, even though the paddle angle for the
Magic and Tenergy rubber would be identical for the same
Dx and Dv, the motion of the paddle would be quite differ-
ent. The necessary fast swing of the Magic paddle along the
direction of a makes juggling the ball with the Magic paddle
more difficult and requires more effort by the player than
with the Tenergy paddle where a slow motion at the correct
angle is sufficient.
Note that in Table I, the experimental difference between
the measured and calculated angles is largest for trials 1 and
5. In both of these cases a is larger than ameasured. A larger a
corresponds to a larger change in the horizontal velocity and
for these two trials the trajectory was the least vertical, with
a horizontal velocity change of 0.3–0.5 m/s. The impact of
the earlier mentioned slight calibration error appears to be
minor, but being successful in getting the ball going straight
up and down impacts the agreement of theory and experi-
mental data more strongly. We think the data presented here
are what one can reasonably expect from students, though
some dedicated students with great hand-eye coordination
may be able to get a lower experimental difference.
Equation (11) restricts all solutions to Dx 3Dv=2r.
However, while all possible solutions must satisfy Eq. (11),
for large angles the normal force decreases and the required
frictional force may exceed the maximum friction that can be
supplied by the paddle rubber. Hence, Eq. (11) is a necessary
but not sufficient condition. We will try in Secs. IV and V to
estimate the required coefficient of friction of the rubber sheet
necessary to exert a large enough frictional force.
IV. TIME OF CONTACT
To observe the impact of the ball, we set the frame rate to
1,000 fps. The camera was placed right next to the paddle
and captured the impact of a ball dropped with little spin
from a height of 0.5–1 m with the paddle placed horizontally
and angled at 45
. We found that the impact for both paddle
angles showed similar results at this temporal resolution.
The high frame rate, and therefore short integration times,
required good lighting conditions.
The images in Fig. 5 were taken using natural sunlight and
the Butterfly paddle placed horizontally. From the video
images, using the size of the ball as a reference, one can esti-
mate the distance of the ball from the paddle. From these dis-
tances and the inspection of the images close to impact, one
can get a rough estimate of the contact time. Of the eight
drops we looked at, two showed contact in only one frame
(like the left sequence in Fig. 5), while two trials showed
contact in two frames (like the right sequence in Fig. 5). The
other four trials had one frame with clear contact and another
frame so close that the ball may or may not have been in
contact with the paddle. The fact that there were trials with
only a single image showing full contact places an upper
limit for the contact time at 2 ms. The trials showing two
sequential images with contact place a lower limit for the
contact time at 1 ms. For contact times of 1–2 ms and typical
changes in speed of 4–8 m/s, Dp=Dt for the 2.7-g ping-pong
ball is in the range of approximately 5–20 N, at least two
orders of magnitude larger than Fg, thus satisfying the condi-
tion that Dp=Dt Fg. The slight downward impulse due to
gravity shown in Fig. 4(c) is therefore indeed negligible. We
can calculate the normal force on the ping-pong ball as
FN ¼ ðDp=DtÞ cos a, and knowing the order of magnitude of
the forces during contact with the paddle, we can investigate
the frictional forces supplied by the paddle.
V. FRICTION
The normal force exerted on the ball by the paddle varies
quickly during impact. For this study, we did not obtain time
resolved force versus time data and we need to make some
simplifying assumptions. We ignore any dependence of the
contact time on the paddle angle, speed, and type, as well as
the speed and spin of the ball. With the rough estimate of
contact time, we can estimate normal and frictional forces if
we know the coefficient of friction of the rubber sheet.
For this, we conducted a classical friction experiment by
sliding the ping-pong ball, with different weights added,
across the surface of interest (the rubber sheet). Figure 6
shows the experimental setup. The ping-pong ball was placed
in a measuring cup and fixed with masking tape so that it
could not rotate. The total mass of the tape, ball, and measur-
ing cup was 61 g. A 50-g weight hanger is attached for all but
the measurement with the lowest normal force. Additional
masses, up to a total of 1,511 g, are added in 100-g incre-
ments. The weight hanger is stabilized with minimal vertical
force with one hand while the other hand applies a horizontal
force that is measured with a force sensor. The force is
increased until the ball starts to slide for a short distance of
1–3 cm, and this is repeated at least six times. The average
and standard deviation of the peak force of six measurements
is calculated and plotted as a function of the normal force in
Fig. 7(a).
For many surfaces, the maximum frictional force increases
linearly with the normal force, with the coefficient of friction
l as the constant of proportionality. However, the elastic
rubber sheet has a coefficient of friction that depends on the
normal force.41
The normal-force-dependent frictional
Fig. 5. High speed capture (1,000 fps) of two drops on the Tenergy paddle.
The ball had little spin and both the paddle and camera were angled horizon-
tally. The images are sequential starting with frames (a/A) and ending with
frames (d/D) with 1 ms between frames. The ball is in contact with the pad-
dle in frame (c) on the left and in frames (B) and (C) on the right.
940 Am. J. Phys., Vol. 84, No. 12, December 2016 Ralf Widenhorn 940

7. coefficient is calculated as the ratio Ff =FN and is plotted in
Fig. 7(b). The data for both rubber sheets can be fitted empir-
ically with a power law of the form
lðFNÞ ¼ 2:1F0:19
N ðTenergyÞ; (15)
and
lðFNÞ ¼ 0:6F0:15
N ðMagicÞ; (16)
where FN is measured in Newtons. For example, for a small
normal force of 1 N the frictional coefficient for the Tenergy
is 2.1, answering the classic physics classroom question if l
can be larger than one. The Tenergy coefficient of friction
decreases for larger normal forces to about 1.3 at FN ¼ 15 N.
For the same normal forces, the Magic rubber sheet has a
coefficient of friction of about 0.6 and 0.4, respectively.
The last column in Table I shows that for trials 1 and 3 the
actual coefficient of friction of the Tenergy rubber vastly
exceeds the lmin values of 0.44 and 0.67. Because of the larger
angle of the Tenergy paddle for trials 4 and 5, the required
lmin values of 1.22 and 1.37 are much closer to the actual fric-
tional coefficient. Meanwhile, for both trials with the Magic
rubber, the coefficients of friction are such that lmin is on the
order of the actual l. The small maximum frictional force is
barely sufficient even for the small angles in these trials, which
is reflected in practice by the difficulty of juggling the ball
with large paddle angles for the Magic rubber. On the other
hand, the larger frictional coefficient of the Tenergy paddle
gives it the feel of more control even for larger angles. The
largest paddle angles can be obtained for a combination of
small Dx; Dv pairs, taking advantage of the higher coefficient
of friction for small normal forces. To accomplish this, one
would need to brush the ball close to the top of its trajectory at
a large angle; this would give the ball a small velocity change,
effectively juggling the ball almost in place.
VI. CONCLUSION
We demonstrated that one can investigate the juggling of a
spinning ping-pong ball with different paddles using basic
concepts from high school or college level introductory phys-
ics and inexpensive and commonly available lab equipment.
A study like this would be an ideal project for students who
enjoy ball sports. Further studies could include the investiga-
tion of contact time, coefficients of restitution, and force, with
higher temporal resolution for different speeds, angles, spins,
paddles, and balls. Moreover, motion analysis could be used
to explore how the speed and direction of the paddle motion
during impact with the ball influences the change in linear and
angular velocities. The activities described here are both well-
defined and rich in interesting open-ended research questions.
The measurements require both experimental skill and appli-
cation of physics that spans most concepts of mechanics in a
way that we hope will be engaging to many students.
ACKNOWLEDGMENTS
The author wants to acknowledge Grace Van Ness,
Michael Fitzgibbons, Pure Pong in the Pearl, and the
anonymous reviewers for their support and helpful feedback.
a)
Electronic mail: ralfw@pdx.edu
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Fig. 6. Setup and free-body diagram for the measurement of the maximum
frictional force for different normal forces.
Fig. 7. (a) Maximum frictional force of the Tenergy and Magic rubber sheets
for different normal forces. (b) Ratio of the maximum frictional force Ff;max
to the normal force FN versus normal force; this ratio represents the effective
coefficient of friction for a particular normal force.
941 Am. J. Phys., Vol. 84, No. 12, December 2016 Ralf Widenhorn 941

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